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Theorem disjsn 3460
Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
disjsn  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )

Proof of Theorem disjsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 disj1 3298 . 2  |-  ( ( A  i^i  { B } )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  { B } ) )
2 con2b 603 . . . 4  |-  ( ( x  e.  A  ->  -.  x  e.  { B } )  <->  ( x  e.  { B }  ->  -.  x  e.  A ) )
3 velsn 3420 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
43imbi1i 231 . . . 4  |-  ( ( x  e.  { B }  ->  -.  x  e.  A )  <->  ( x  =  B  ->  -.  x  e.  A ) )
5 imnan 634 . . . 4  |-  ( ( x  =  B  ->  -.  x  e.  A
)  <->  -.  ( x  =  B  /\  x  e.  A ) )
62, 4, 53bitri 199 . . 3  |-  ( ( x  e.  A  ->  -.  x  e.  { B } )  <->  -.  (
x  =  B  /\  x  e.  A )
)
76albii 1375 . 2  |-  ( A. x ( x  e.  A  ->  -.  x  e.  { B } )  <->  A. x  -.  (
x  =  B  /\  x  e.  A )
)
8 alnex 1404 . . 3  |-  ( A. x  -.  ( x  =  B  /\  x  e.  A )  <->  -.  E. x
( x  =  B  /\  x  e.  A
) )
9 df-clel 2052 . . 3  |-  ( B  e.  A  <->  E. x
( x  =  B  /\  x  e.  A
) )
108, 9xchbinxr 618 . 2  |-  ( A. x  -.  ( x  =  B  /\  x  e.  A )  <->  -.  B  e.  A )
111, 7, 103bitri 199 1  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409    i^i cin 2944   (/)c0 3252   {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2948  df-in 2952  df-nul 3253  df-sn 3409
This theorem is referenced by:  disjsn2  3461  orddisj  4298  ndmima  4730  funtpg  4978  fnunsn  5034  ressnop0  5372  ftpg  5375  fsnunf  5390  fsnunfv  5391  phpm  6358  fiunsnnn  6369  ac6sfi  6383  fzpreddisj  9035  fzp1disj  9044  frecfzennn  9367
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